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A&A
Volume 686, June 2024
Article Number A179
Number of page(s) 9
Section Celestial mechanics and astrometry
DOI https://doi.org/10.1051/0004-6361/202347738
Published online 11 June 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

Continuous tracking of stellar orbits in the Galactic centre over the past few decades revealed the massive compact object Sagittarius A* (Sgr A*) at their shared focal point (Nobel Committee for Physics 2020; Genzel 2022). These and other observations, including those of flare orbits (GRAVITY Collaboration 2018b, 2023a; Wielgus et al. 2022) and the black hole shadow (Event Horizon Telescope Collaboration 2022), are in best agreement with Sgr A* being a massive black hole of about 4 million solar masses, and they rule out many alternatives. Assuming the black hole model, the orbit of the star S2 (also S0-2) further allowed observations of relativistic effects, namely of the gravitational redshift (GRAVITY Collaboration 2018a; Do et al. 2019; Saida et al. 2019) and of the Schwarzschild precession (GRAVITY Collaboration 2020).

While there is currently no visible matter between the innermost known S-stars and the accretion flow of Sgr A*, observations still allow for the presence of both compact and distributed objects in this region, though with strong constraints. Compact matter could, for example, be an intermediate-mass black hole (Tep et al. 2021; GRAVITY Collaboration 2023b; Evans et al. 2023; Will et al. 2023). Distributed matter could be a cluster of faint stars and stellar remnants (Jiang & Lin 1985; Rubilar & Eckart 2001; Merritt 2013), dark matter (Gondolo & Silk 1999; Gnedin & Primack 2004; Sadeghian et al. 2013; Brito et al. 2020), or a combination thereof. Observations currently allow for a few thousand solar masses of distributed matter within a ball of the radius of the apocentre distance of S2 (GRAVITY Collaboration 2022), and this upper bound (or detection threshold) is predicted to decrease significantly in the coming years as the apocentre half of the orbit is being observed with the currently most sensitive instruments (Heißel et al. 2022).

Depending on the nature of the distributed matter, theory predicts it to attain certain density profiles. On their full scale, nuclear stellar clusters have been shown to relax to a Bahcall-Wolf cusp (Bahcall & Wolf 1976), that is, to a radial density power law with exponent −7/4. However, in the context of the innermost Galactic centre, nuclear clusters are also often modelled as Plummer distributions (Plummer 1911) in order to account for a realistically finite core density. Popular candidates for cold dark matter around a massive black hole, on the other hand, are predicted to attain a power-law cusp density distribution with exponents in the range from 0.5 to 2.5 constrained by radio and gamma-ray observations; this transitions to a core plateau depending on the rate of particle self-annihilation (Gondolo & Silk 1999; Bertone et al. 2002; Gnedin & Primack 2004; Sadeghian et al. 2013; Fields et al. 2014; Shapiro & Shelton 2016; Chan 2018; Balaji et al. 2023; Zuriaga-Puig et al. 2023). The central ‘spike’ of such a cusp can further be softened by the scattering of the dark matter by a cluster of faint stars and stellar remnants due to both dynamical heating and capture by the black hole (Merritt 2004; Bertone & Merritt 2005) or as a consequence of a history of hierarchical mergers of dark matter halos containing massive black holes (Merritt et al. 2002). Other forms of dark matter again predict different profiles (see, e.g. Detweiler 1980, Cardoso & Yoshida 2005, Dolan 2007, Witek et al. 2013, and Brito et al. 2020 for scalar fields, including types of fuzzy cold dark matter Cardoso et al. 2022).

Because of this large variety of dark matter candidates and models, not all of which are mutually exclusive (as already noted, a combination of a stellar cluster and dark matter is plausible), it is difficult to obtain general constraints concerning the amount, distribution, and nature of dark matter that might reside in the Galactic centre. Results such as the quoted current upper observational bound from stellar orbits of a few thousand solar masses come with a prior assumption of the radial density profile of the distribution, and they are thus model dependent. In this way, statements about the possible nature of the dark matter rely on an incomplete comparison between a zoo of models.

In this work, we present a model-agnostic approach based on an approximation of dark matter profiles by a series of concentric spherical mass shells. The model parameters are the shell masses that are estimated by fitting to observational data. This idea is also used in the mapping of mass concentrations (mascons) for celestial bodies via the geodesy of orbiting spacecrafts (Werner & Scheeres 1996; Izzo & Gómez 2022). The advantage of the mass shell model is that a fit not only estimates the amount but also the radial profile of the underlying density distribution without any priors beyond, in this case, sphericity. Consequently, constraints such as observational upper bounds on total dark matter mass can be given model independently, and a single model fit can, in principle, discriminate among various physically reasonable dark matter models and compositions. In what follows, we illustrate this through the particular example of S2.

In Sec. 2, we present our model for a star orbiting a massive black hole through a dark matter distribution and discuss our model and fitting implementation. In Sect. 3, we use our model to analyse mock observations of S2, focusing on the degree to which the amount and radial profile of distributed matter are constrained by the data. We conclude with a discussion on the implications of our results for future observational dark matter constraints in Sec. 4.

2 Methods

2.1 Equations of motion

We consider the scenario of a star (S2) orbiting a massive black hole (Sgr A*) through a distribution of dark matter. The near-Keplerian orbits of S2 allow us to formulate the problem by the equations of motion: r¨=−GM•r2n+a1PN+aDM,$\[\ddot{\mathbf{r}}=-\frac{G M_{\bullet}}{r^2} \mathbf{n}+\mathbf{a}_{\mathrm{IPN}}+\mathbf{a}_{\mathrm{DM}},\]$(1)

where r is the position of S2, r = |r|, n = r/r, G is the gravitational constant, M• is the black hole mass, and a dot denotes time derivation (Merritt 2013; Poisson & Will 2014; Heißel et al. 2022). The first term describes the extreme mass ratio Kepler problem for S2 and Sgr A*. a1PN takes into account relativity for the two-body problem to first post-Newtonian order (sufficient for S2 with a pericentre distance of about 1400 Schwarzschild radii). For our extreme mass ratio case, in harmonic coordi-nates1, it is given by a1PN=4GM•c2r2((GM•r−|v|24)n+(n⋅v)v),$\[\mathbf{a}_{1 \mathrm{PN}}=4 \frac{G M_{\bullet}}{c^2 r^2}\left(\left(\frac{G M_{\bullet}}{r}-\frac{|\mathbf{v}|^2}{4}\right) \mathbf{n}+(\mathbf{n} \cdot \mathbf{v}) \mathbf{v}\right),\]$(2)

where v=r˙$\[\mathbf{v}=\dot{\mathbf{r}}\]$ and c is the speed of light (Merritt 2013; Poisson & Will 2014). We neglect the very small effects due to black hole spin (Will 2008; Zhang et al. 2015; Yu et al. 2016; Grould et al. 2017; Waisberg et al. 2018; Qi et al. 2021; Alush & Stone 2022; Capuzzo-Dolcetta & Sadun-Bordoni 2023). Here, aDM denotes the acceleration exerted by the dark matter, which we assume to be sufficiently small for a Newtonian description of aDM to be adequate (Heißel et al. 2022, Appendix B). Furthermore, we assume the dark matter distribution to be spherically symmetric and centred at Sgr A*.

2.2 Shell model

In order to give our dark matter model the flexibility to attain a wide variety of physically reasonable radial profiles, we construct it as a sum of concentric shells of radii (ri)i=1N$\[\left(r_i\right)_{i=1}^N\]$ and masses (mi)i=1N$\[\left(m_i\right)_{i=1}^N\]$. The total mass of the distribution enclosed within a ball of a certain radius MC (r) then increases in steps with r, such that by Newton’s shell theorem aDM(r)=−GMC(r)r2n,$\[\mathbf{a}_{\mathrm{DM}}(r)=-\frac{G M_C(r)}{r^2} \mathbf{n},\]$(3)

with MC(r)=∑i=1Nmi1+f(r−ri)2. $\[M_C(r)=\sum_{i=1}^N m_i \frac{1+f\left(r-r_i\right)}{2} \text {. }\]$(4)

Thin (radially Dirac density distributed) shells correspond to equating f to the Heaviside step function. For differentiability and a better adaptability to physically reasonable profiles, we, however, smooth out the steps to sigmoids given by the logistic function f(r − ri) = tan h((r − ri)/κ), where κ controls the smoothness of the increments. Calculating MC′(r)=∑i=1N(2κ)−1mi$\[M_C^{\prime}(r)=\sum_{i=1}^N(2 \kappa)^{-1} m_i\]$ sech((r − r0)/κ)2, it follows that each dark matter shell has a radial density distribution resembling a bell shape of width κ, marking a roughly 65% drop-off. A closely related alternative would be to use the error function for f, in which case the dark matter shells would have Gaussian radial density distributions.

2.3 Flexibility of the shell model

Let us denote the enclosed mass function of a given ground truth (GT) distribution by MCGT(r)$\[M_C^{\mathrm{GT}}(r)\]$. Then, by choosing an equal spacing Δr between the ri, we define the shell approximation of the ground truth distribution by Eq. (4) with m1=MCGT(r1+Δr/2)$\[m_1=M_C^{\mathrm{GT}}\left(r_1+\Delta r / 2\right)\]$ and mi+1=MCGT(ri+1+Δr/2)−mi$\[m_{i+1}=M_C^{\mathrm{GT}}\left(r_{i+1}+\Delta r / 2\right)-m_i\]$, which is similar to a midpoint Riemann sum.

In order to demonstrate the flexibility of our model, we consider the following three examples: MCGT(r)∝{(rr0)5/4 Bahcall-Wolf cusp (rr0)3(1+r2r02)−3/2 Plummer model (rr~0)11(1+r10r~010)−11/10 Zhao model, α=110,$\[M_C^{\mathrm{GT}}(r) \propto\left\{\begin{array}{ll}\left(\frac{r}{r_0}\right)^{5 / 4} & \text { Bahcall-Wolf cusp } \\\left(\frac{r}{r_0}\right)^3\left(1+\frac{r^2}{r_0^2}\right)^{-3 / 2} & \text { Plummer model } \\\left(\frac{r}{\tilde{r}_0}\right)^{11}\left(1+\frac{r^{10}}{\tilde{r}_0^{10}}\right)^{-11 / 10} & \text { Zhao model, } \alpha=\frac{1}{10}\end{array},\right.\]$(5)

where r0 = 2483.1 AU (0.3″ on sky) and r~0$\[\tilde{r}_0\]$ = 1200 AU (0.15″ on sky). All profiles are normalised such that MCGT(ra)=10−3M•$\[M_C^{\mathrm{GT}}\left(r_{\mathrm{a}}\right)=10^{-3} M_{\bullet}\]$, where ra is the apocentre distance of S2, which corresponds to the current observational upper bound of GRAVITY Collaboration (2022). The distribution of Bahcall & Wolf (1976) resembles a dynamically relaxed nuclear star cluster, but it also serves as a cold dark matter spike example (without particle self-annihilation) since its power-law exponent lies well within the observational constraints for the respective models, which are also power laws (Gondolo & Silk 1999; Gnedin & Primack 2004; Fields et al. 2014; Sadeghian et al. 2013). The profile by Plummer (1911) represents a relaxed stellar cluster, which is, however, also often used to model the innermost region of nuclear clusters in order to account for a realistically finite central density. The Zhao family of double power-law distributions Zhao (1996, Table 1,) provides a parametrisation that is flexible enough to capture a variety of analytical DM halo models, and for certain parameter ranges has shown excellent agreement with DM simulations. We restrict our attention to a particular instance of this family, the so-called α-model with α = 1/10, as this leads to a profile that differs qualitatively from that of Plummer and Bahcall-Wolf.

Figure 1 shows two approximations with five mass shells to the above ground truth models. In the κ = 6 approximation, the individual mass shell distributions are clearly separated resulting in well-pronounced, enclosed mass steps. In the κ = 2 approximation, the mass shells have more overlap and the enclosed mass function increases smoothly, resulting in a better approximation of the example profiles; the errors are smaller than 5 × 10−3 (Bahcall-Wolf), 8 × 10−3 (Plummer), and 2 × 10−2 (Zhao) percent of M•.

thumbnail Fig. 1

Approximations with five mass shells (orange) for κ = 6 (top) and κ = 2 (bottom) of three different ground truth profiles: Bahcall-Wolf (left), Plummer (centre), Zhao (right). Solid vertical lines mark the centre positions of the shells ri. Dashed lines mark the pericentre and apocentre of S2.

2.4 Orbit model

Equations (1)–(4) together complete our model for the physical stellar orbit. The apparent orbit in terms of the observables’ right ascension (RA), declination (DEC), and radial velocity (RV) was obtained by projecting the physical orbit onto the coordinate system of the observer (see, e.g. Heißel et al. 2022, Fig. 1 and Eqs. (A.7) and (A.8)). The transformation from physical to angular distances thereby introduces the distance of the observer on Earth to the Galactic centre R0 as an additional parameter.

In summary, our model produces the three observables RA, DEC, RV at given observation times and for given model parameters. Of these parameters, we always fix the (equally spaced) shell positions ri and the shell widths κ. For simplicity, we also fix the black hole mass M• and the distance to the Galactic centre R0. This leaves our model with 6 + N free parameters to be estimated by fitting the following to the data: six initial conditions for Eq. (1) and the N shell masses mi of Eq. (4), which enter the equations of motion (Eq. (1)) via the dark matter acceleration term (Eq. (3)). For all our case studies of Sect. 3, we choose N = 5 and κ = 2 for the creation of both the mock observations and the fitting of the model to these data.

2.5 Model simplifications

For simplicity, we neglect observational effects such as the Rømer delay, the gravitational redshift, the transverse Doppler shift, and the motion of the Solar System. While these are observable and thus crucial to model when fitting data of real observations (Zucker et al. 2006; Grould et al. 2017; Alexander 2017; GRAVITY Collaboration 2018a; Do et al. 2019; Saida et al. 2019), they would, unnecessarily, complicate the model for the mock data analysis of Sect. 3, in which we focus on whether it is possible to discriminate between dark matter models in principle. This choice does not affect our analysis since it does not result in a reduction of model parameters, and since the dark matter affects the orbit in precisely the same way, regardless of whether the effects are modelled or not. We verified this first by comparing the residual orbits between models with and without dark matter in two cases: when we model these observational effects and when we do not; the residual orbits of both cases overlap. Second, we perform 250 Markov chain Monte Carlo (MCMC) fits to mock data, once modelling the observational effects for both the creation of the data and the fitting, and once neglecting them; both cases produce the same statistics in estimated model parameter values.

The fixing of the black hole mass M• and the distance to the Galactic centre R0 on the other hand results in a reduction of model parameters, and thus it has to be taken into account when interpreting the results of our analysis. We do this in Sect. 4.1.

Table 1

Ground truth and initial guess.

2.6 Model implementation and fitting algorithm

Instead of integrating Eq. (1) directly, we use the equivalent system of osculating equations in the form of Poisson & Will (2014, Eqs. (3.64)–(3.66),) to obtain our model orbits. These are a system of six evolution equations in time for the following set of osculating orbital elements: the semi-latus rectum p, the eccentricity e, the inclination ι, the argument of pericentre ω, the argument of the ascending node Ω, and the true anomaly f of the osculating orbits. We denote the initial elements by a subscript 0. The transformation law from the position and velocity to the elements is given by Poisson & Will (2014, Eqs. (3.40) and (3.41)). For the integration, we utilise the open-source higher order Taylor integrator heyoka by Biscani & Izzo (2021). To fit our model to data, we use gradient descent of least squares weighted by the measurement uncertainties. For this, we use the Adam optimiser by Kingma & Ba (2014) with the recommended default parameters and a learning rate of 10−5. The gradient of the model function with respect to the model parameters (i.e. the model sensitivities) is given by variational equations, which are simultaneously integrated with the model function. In each iteration, the model error is calculated and the initial guesses for both the initial conditions and the shell masses are improved in the negative direction of the gradient.

thumbnail Fig. 2

Reconstructed enclosed mass curves corresponding to best fits to 300 mock observations with vanishing instrument uncertainties. The three sets of mock data are created with the mass shell approximations (solid lines) of the profiles of Eq. (5). The fits are initiated from IG1 (Table 1).

3 Results

3.1 Ideal data: Vanishing measurement uncertainties

For the ground truth, which generates our mock observations, we take our model of Sect. 2.4 with the (GT) parameter values of Table 1 and the dark matter profiles of Eq. (5) approximated by five mass shells with κ = 2 (Sect. 2.3 and Figs. 1d–f).

We start our investigation with 300 (RA, Dec, RV) mock observations that are equally time distributed over one orbital period of S2. This is a conservative estimate, taking into account down time for commissioning (among other things) for the amount of data that will be gathered if the current observing rate with the Very Large Telescope (VLT) GRAVITY interferometer (GRAVITY Collaboration 2017) and ERIS spectrometer (Davies et al. 2018b) continues. We then perform fits as described in Sect. 2.6. In the idealised scenario of vanishing instrument uncertainties and starting from the initial guess IG1 of Table 1, our model is able to reconstruct the three ground truth distributions from the fits up to machine precision (Fig. 2). This is proof of principle, whereby we initialise mascon masses to vanish; hence, we assume no prior information on the underlying distribution and fit a model flexible enough to have local optima away from the ground truth and nevertheless attain it.

3.2 Idealised data: Measurement uncertainties of 1/10th of the current instrument precision

Introducing Gaussian noise to the observable (RA, Dec, RV) data to simulate measurement uncertainties, we find that the reconstructed enclosed mass profiles now vary depending on both the noise realisation of the data (i.e. on the specific data instance for the observables drawn from the statistical distribution) and the initial parameter guess. We also find that a covariance analysis yields small uncertainties in the estimated shell masses. Taken together, these points suggest that the problem is not convex such that the least squares map now indeed exhibits multiple local minima to which the algorithm can descend, or narrow valleys in which it stagnates. One likely reason for this is the degeneracies between neighbouring mass shells; an overestimated mi can, within measurement uncertainties, be balanced by an underestimated mi+1. To monitor this statistically, we perform a sample of ten fits per ground truth, where with each fit we vary not only the noise realisation (i.e. for each fit we draw new (RA, Dec, RV) samples from Gaussian distributions), but also the initial guess for the parameters within reasonable bounds. For these bounds, we orient ourselves towards the current observational constraints of GRAVITY Collaboration (2022, Table B.1,) and draw the initial orbital elements from the normal distributions IG2 or IG3 of Table 1, depending on the chosen measurement uncertainties. On the other hand, we set the shell masses to be equal to zero when initiating the fits. While the factor ten difference between the standard deviations of IG2 and IG3 is chosen ad hoc, the trend is motivated by the fact that smaller measurement uncertainties would also yield narrower bounds for the parameter estimates.

Choosing measurement uncertainties of one tenth of the current instrument performances in measuring S2, that is, 5μ″ in astrometry and 1 km s−1 in RV (GRAVITY Collaboration 2020, 2022), and drawing the initial guess from IG2, we obtain Fig. 3a. We see that while even for this low noise level the reconstructed enclosed mass profiles spread, they still cluster around the correct respective ground truths. In particular, for low-to-mid radii the model is able to clearly distinguish between the three profiles. For large radii, however, the fits are less constrained, in particular for radii greater than about 1600 AU – the domain of the outermost shell. Consequently, the overall amount of distributed matter within the apocentre of S2 is poorly constrained. We interpret this as follows. Shells located at lower radii influence the orbit over longer timescales, and they are thus constrained by more data over time than shells located at larger radii. From Fig. 4, we see that the initial osculating orbital elements converge mostly within the statistics from which they were initially drawn. We also observe two correlations. A larger initial eccentricity preferably pairs with a smaller initial semi-latus rectum, which can be understood from the fact that p ∝ (1 − e2). The fact that a larger initial true anomaly preferably pairs with a smaller initial argument of pericentre follows directly from the geometric relation between the two angles (see, e.g. Heißel et al. (2022, Fig. 1)). The respective plots for the orbital elements for all the cases that follow look qualitatively the same as in this case.

Increasing the number of observations to 3000 while keeping them confined within a single orbit and repeating the above procedure results in Fig. 3c. We see that the tenfold increase in observations only yields an insignificant narrowing of the spread of the reconstructions in the low-to-mid radii regime, and no improvement in the large radii regime, including the estimation of the overall amount of distributed mass within the apocentre. In contrast, spreading the 3000 observations over ten orbital periods, we see a strong improvement in the spread of the reconstructions for all radii (Fig. 3e). Both the Bahcall-Wolf and the Plummer GT profiles are reconstructed unambiguously; however, the fits failed to reconstruct the Zhao profile. Redoing this case with an initial guess for the mi close to their GT values does recover the Zhao profile. We conclude that in the Zhao case, the gradient descent, initiated from mi = 0, becomes trapped in a local optimum on the least-squares map, with a value around that of the GT optimum. Remarkably, however, even though the profile of the Zhao distribution is not found when initiating from mi = 0, the estimate on the overall enclosed mass within the apocentre is still precise and accurate.

We can understand this when we think of the orbital dynamics in terms of its secular and non-secular components (Merritt 2013). The former are changes in the orbital elements that accumulate per orbit. The latter are changes in the orbital elements that average to zero per orbit. In our present case, there is only one secular effect on the orbit caused by the dark matter, which is a (negative) pericentre advance ∂t⟨ω⟩t, adding a retrograde component to the precession of the orbit within its plane (Jiang & Lin 1985; Merritt 2013; Rubilar & Eckart 2001; Heißel et al. 2022), a prograde component is caused by the first post-Newtonian correction (Merritt 2013; Poisson & Will 2014; GRAVITY Collaboration 2020). The non-secular effects caused by the distributed matter correspond to the small variations in the orbital elements over time and consequently to the small distinct distortions from a Kepler ellipse, which the different profiles inscribe into each orbit. In the case of Figs. 3a and c the measurement uncertainty is sufficient to capture the non-secular effects for low and mid radii, and therefore it becomes possible to differ between the signatures that each GT profile inscribes into the single orbit. The lack of data over multiple orbits, however, prevents good constraints on the pericentre advance per orbit, and therefore on the overall enclosed mass within the apocentre. In the case of Fig. 3e, on the other hand, the data over multiple orbits allow good constraints on both profile and amount, in two out of three cases. In the Zhao case, on the other hand, a good fit was found with a local optimum corresponding to a profile distinct from that of the GT. This shows that different profiles, and hence different, non-secular signatures, are compatible with the same data within measurement uncertainties. Since these measurement uncertainties have so far been underestimated, this strongly suggests that an unambiguous constraining of the radial profile of an underlying distribution by future observations with the current and next generation instruments is not possible. What appears very feasible with future data, however, is putting unbiased constraints on the overall amount of enclosed mass within the apocentre, that is, constraints that are not subject to an a priori assumption on the functional form of the distribution. The fact that the amount of mass is estimated well even in the Zhao case of Fig. 3f lends credence to this conclusion, as do the results that follow for realistic measurement uncertainties.

3.3 Realistic data: Measurement uncertainties of the current instrument precision

The right panel of Fig. 3 shows the results for the same sequence of cases as in the left panel (Sec. 3.2), but with realistic measurement uncertainties in the mock data of 50μ″ and 10 km s−1, that is, with the current performances of GRAVITY, SINFONI, and ERIS in tracking S2. Clearly, with data confined to one orbit, the fits fail to reconstruct any of the ground truth distributions, and there are also no statistical trends, not even for low radii (Figs. 3b,d). The measurement uncertainties are simply too large to capture the non-secular variations within a single orbit. For 3000 observations over ten, orbits clear trends again emerge, such that the Plummer and Bahcal-Wolf profiles can be reconstructed unambiguously, though with a significantly larger spread, as in the case of Fig. 3e. The fits, however, still fail to reconstruct the profile of the Zhao GT and find another locally optimal profile instead. The enclosed mass within the apocentre is again well constrained for all cases, including the Zhao model, given the increased measurement uncertainty. This indicates that future observations over several orbits will indeed allow for unbiased constraints on the overall amount of enclosed mass within the apocentre.

thumbnail Fig. 3

Statistics of reconstructed enclosed mass curves resulting from ten fits per ground truth case (Eq. (5)) to different mock data. Left: Measurement uncertainties of 5μ″, 1 km s−1, and initial guess drawn from IC2 of Table 1. Right: Measurement uncertainties of 50μ″, 10 km s−1, and initial guess drawn from IC3 of Table 1 (right). Top: 300 mock observations over one orbital period. Middle: 3000 mock observations over one orbital period. Bottom: 3000 mock observations over ten orbital periods. Also shown are the means (dashed), standard deviations (shaded), and shell approximated ground truth profiles (solid).
thumbnail Fig. 4

Best-fit values (dots) for initial osculating orbital elements corresponding to cases of Fig. 3a. The cross in each plot marks the mean and standard deviation of the normal distribution IC2 from which the initial guesses for the fits have been drawn. The corresponding plots for all other considered cases look qualitatively the same.
thumbnail Fig. 5

Shell mass statistics corresponding to Bahcall-Wolf case of Fig. 3f. Care is to be taken when reading this plot, as the shells are not thin (Dirac distributed) but smoothed out (Sec. 2.2). Here, the shell masses are displayed at the radius at which the respective shell is centred.

3.4 Comparison to the shell mass profile

In the presentation of the above results, we chose plots of the enclosed mass profiles over plots of the corresponding shell masses. While the latter presentation would show the estimated shell model parameters directly, it is also less intuitive to infer from it the radial DM profiles, since the shells are not thin (that is, Dirac distributed), but smoothed out (Sec. 2.2). Another reason why we preferred the presentation in terms of enclosed mass is that it is the enclosed mass that directly enters the equations of motion and is thus the quantity directly constrained by the data. As an example, we show, in Fig. 5, the best-fit shell mass statistics corresponding to the Bahcall-Wolf case of Fig. 3f.

4 Conclusions

4.1 Summary

In this paper, we present a novel approach to modelling spherically symmetric matter distributions by concentric mass shells. We implemented this concept in a relativistic model for a star (S2) orbiting a massive black hole (Sgr A*) through a dark matter distribution (cluster of faint stars and stellar remnants, dark matter, or a combination thereof). Via mock data experiments, we then arrived at the following key results.

  1. We performed a proof of principle that, given astrometric and spectroscopic observations of sufficient precision, a fit of our model is able to find an underlying ground truth dark matter distribution with enough accuracy to discriminate between various physical models and compositions. Unlike with conventional models based on certain functional forms for the density distribution, no a priori assumption about the principal profile of the ground truth is made.

  2. Using the spherical shell model, our results indicate that with the measurement uncertainty of current and next-generation instruments it will not be possible to achieve unbiased constraints on the radial profile of an underlying matter distribution.

  3. We showed, however, that future observations can yield unbiased constraints on the overall amount of enclosed mass within the apocentre, though such constraints require observations over multiple orbital periods. We interpreted these results from the theoretical perspective of secular versus non-secular orbital dynamics, where a capturing of the former is required for good constraints on the overall amount of mass within the apocentre; however, in addition a capturing of the latter is required to yield unambiguous constraints on the radial profile.

As mentioned in Sec. 2.5, these results were obtained in a model with fixed parameter values for the black hole mass M• and the distance to the Galactic centre R0. If these two parameters were also to be fitted, the uncertainties in the other estimated parameter values would be expected to increase. Nevertheless, Result 1 would still hold, since there are no measurement uncertainties in this case. Result 2 would also still hold, since the claim is only strengthened in a more complicated model (i.e. with more parameters to fit). Result 3 is, however, affected by the simplification, in that the full model would likely require more observations than the simplified model to yield the same constraints. Yet, the basic message, that such unbiased constraints can be achieved given enough observations over a sufficient number of orbits, is still expected to hold.

4.2 Outlook

Concerning the prospects to improve the measurement uncertainties, it is important to note that the uncertainty in RV is more constrained by the source S2 (that is, by the width of the measured spectral line) than by the instrument, such that ERIS does not bring significant improvement compared to SINFONI (Eisenhauer et al. 2003; Bonnet et al. 2004). While the GRAVITY+ upgrade (Eisenhauer 2019) will drastically improve the sensitivity of the instrument with respect to fainter objects, a significant reduction of the astrometric uncertainties for S2 is not expected. In summary, the 50μ″ and 10 km s−1 uncertainties we assumed will remain realistic in the upcoming years.

Since our above discussion was limited to the use of data for a single star (S2), it is reasonable to ask if the use of data for more stars would yield different conclusions. This is of particular interest regarding the development of the Extremely Large Telescope (ELT). With its Multi-AO Imaging Camera for Deep Observations (MICADO; Davies et al. 2018a), it is expected to track the S-stars with similar uncertainties to GRAVITY (Pott et al. 2018). Its much wider field of view will, however, allow it to observe a whole family of S-stars in a single frame, thus enabling it to gather data for more stars at a much higher rate than is currently possible. Nevertheless, assuming spherical symmetry, we do not expect to arrive at different conclusions than in our present discussion when incorporating data for more stars. This is because our results indicate that the limiting factor to capture the non-secular effects, and thus the radial profiles, is the measurement accuracy, which data from more stars do not improve. For the constraints on the secular effect and thus on the overall amount of distributed mass within the apocentre, we showed that the limiting factor is the number of observed periods; hence, in this respect data for more stars are not helpful. It is true, though, that different stellar orbits probe different radial domains, and they therefore constrain the distribution slightly better in different regions. Also, more eccentric orbits are more susceptible to the non-secular component of the dynamics. However, the true benefit of multiple stellar data would only unfold if one drops the assumption of spherical symmetry, since in that case different orbital orientations become important.

In addition to improving the unbiased constraints on the overall amount of distributed matter within the apocentre, future observations of the S-stars will certainly continue to improve biased constraints on specific classes of dark matter distributions – a topic that has seen a strong and very recent rise in interest (Fujita & Cardoso 2017; Ferreira et al. 2017; Lacroix 2018; Bar et al. 2019; GRAVITY Collaboration 2019, 2022; Becerra-Vergara et al. 2020, 2021b,a; Argüelles et al. 2023; Heißel et al. 2022; Chan et al. 2022; Yuan et al. 2022; Bambhaniya et al. 2022; Della Monica & de Martino 2022, 2023b,a; Chan & Lee 2023; Shen et al. 2024). Our results, however, underline the problem that this approach faces: its results all come with their own and distinct a priori assumptions, and thus fail to be mutually exclusive. In other words, the biased approach does not contribute to a downsizing of the zoo of candidate models, but rather helps to restrict the parameter spaces of single candidate models. Despite being important, this task is of limited utility for answering the bigger question about the underlying nature of dark matter.

The potential to constrain dark matter in a model-agnostic manner, which we investigated in this work, is one of several examples for the physics that we can learn from precise observations of stellar orbital perturbations in the Galactic centre; on secular timescales, these manifest themselves as precessions. For example, the spin of the central black hole or the perturbation stemming from an intermediate-mass black hole (IMBH) as well as stellar perturbations would likewise cause orbital precessions. Since the magnitudes of these effects can compete very well, the question of how different precessions can be disentangled when analysing observational data will be key as these regimes become observable. A prime example is the necessity of disentangling the contributions of the black hole spin and quadrupole moment (i.e. its oblateness) in order to test the no-hair theorem (Will 2008; Merritt et al. 2010; Qi et al. 2021). A recent suggestion for the disentangling of secular precessions exploits the fact that orbits that precess on secular timescales (i.e. over multiple periods) for several causes, still entail the distinct signatures of each of these causes on non-secular timescales (i.e. within one period; Heißel et al. 2022; Alush & Stone 2022). Although these and other important insights exist (see, e.g. Merritt et al. (2010) for a discussion specific to stellar perturbations), the very important question of disentangling precessions is perhaps not yet sufficiently covered by the literature.

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1

A recent discussion on the PN expansion in different coordinates is given in Ribeiro et al. (in prep.). For a closely related discussion on coordinate effects in relativistic orbit models, see Heißel et al. (in prep.).

All Tables

Table 1

Ground truth and initial guess.

In the text

All Figures

thumbnail Fig. 1

Approximations with five mass shells (orange) for κ = 6 (top) and κ = 2 (bottom) of three different ground truth profiles: Bahcall-Wolf (left), Plummer (centre), Zhao (right). Solid vertical lines mark the centre positions of the shells ri. Dashed lines mark the pericentre and apocentre of S2.
In the text
thumbnail Fig. 2

Reconstructed enclosed mass curves corresponding to best fits to 300 mock observations with vanishing instrument uncertainties. The three sets of mock data are created with the mass shell approximations (solid lines) of the profiles of Eq. (5). The fits are initiated from IG1 (Table 1).
In the text
thumbnail Fig. 3

Statistics of reconstructed enclosed mass curves resulting from ten fits per ground truth case (Eq. (5)) to different mock data. Left: Measurement uncertainties of 5μ″, 1 km s−1, and initial guess drawn from IC2 of Table 1. Right: Measurement uncertainties of 50μ″, 10 km s−1, and initial guess drawn from IC3 of Table 1 (right). Top: 300 mock observations over one orbital period. Middle: 3000 mock observations over one orbital period. Bottom: 3000 mock observations over ten orbital periods. Also shown are the means (dashed), standard deviations (shaded), and shell approximated ground truth profiles (solid).
In the text
thumbnail Fig. 4

Best-fit values (dots) for initial osculating orbital elements corresponding to cases of Fig. 3a. The cross in each plot marks the mean and standard deviation of the normal distribution IC2 from which the initial guesses for the fits have been drawn. The corresponding plots for all other considered cases look qualitatively the same.
In the text
thumbnail Fig. 5

Shell mass statistics corresponding to Bahcall-Wolf case of Fig. 3f. Care is to be taken when reading this plot, as the shells are not thin (Dirac distributed) but smoothed out (Sec. 2.2). Here, the shell masses are displayed at the radius at which the respective shell is centred.
In the text

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